Given a complex Hilbert space H, we study the manifold $$𝒜$$ of algebraic elements in $$Z=ℒ\left(H\right)$$ . We represent $$𝒜$$ as a disjoint union of closed connected subsets M of Z each of which is an orbit under the action of G, the group of all C*-algebra automorphisms of Z. Those orbits M consisting of hermitian algebraic elements with a fixed finite rank r, (0< r<∞) are real-analytic direct submanifolds of Z. Using the C*-algebra structure of Z, a Banach-manifold structure and a G-invariant torsionfree affine...

The notion of a bilattice was introduced by Shulman. A bilattice is a subspace analogue for a lattice. In this work the definition of hyperreflexivity for bilattices is given and studied. We give some general results concerning this notion. To a given lattice $ℒ$ we can construct the bilattice ${\Sigma }_{ℒ}$. Similarly, having a bilattice $\Sigma$ we may consider the lattice ${ℒ}_{\Sigma }$. In this paper we study the relationship between hyperreflexivity of subspace lattices and of their associated bilattices. Some examples of hyperreflexive...

We study reflexivity of bilattices. Some examples of reflexive and non-reflexive bilattices are given. With a given subspace lattice $ℒ$ we may associate a bilattice ${\Sigma }_{ℒ}$. Similarly, having a bilattice $\Sigma$ we may construct a subspace lattice ${ℒ}_{\Sigma }$. Connections between reflexivity of subspace lattices and associated bilattices are investigated. It is also shown that the direct sum of any two bilattices is never reflexive.